1. Prepare Soup: Tie half of dill and half of parsley in a bunch with kitchen string. Chop remaining dill and parsley to equal 2 tsp. each.
2. Bring chicken, next 4 ingredients, dill-parsley bunch, and 3 1/2 qt. water to a boil in a large Dutch oven over medium-high heat; skim any foam with a slotted spoon. Cover, reduce heat to medium-low, and simmer, 2 1/2 to 3 hours or until chicken is tender and falls off the bone.
3. Meanwhile, prepare Matzo Balls batter: Whisk together oil, eggs, and 1/4 cup water. Add matzo meal and 1 3/4 tsp. kosher salt; whisk until well blended. Cover and chill 30 minutes.
4. Remove soup from heat. Skim fat from surface of broth. Remove chicken and celery. Pour broth through a fine wire-mesh strainer into a large bowl. Return broth, carrots, and parsnips to Dutch oven, discarding onion and herb bunch. Let chicken, celery, and broth mixture cool 30 minutes.
5. Meanwhile, assemble and cook Matzo Balls: Shape matzo batter into 18 (1-inch) balls (about 1 Tbsp. each), using wet hands. Bring 2 1/2 qt. water to a boil in a large saucepan over medium-high heat. Drop matzo balls into boiling water; return to a boil. Cover, reduce heat to medium-low; simmer 30 minutes. Remove matzo balls from water with a slotted spoon.
6. Squeeze juice from cooled celery ribs into broth. Discard celery ribs. Skin and bone chicken; shred chicken. Add matzo balls, shredded chicken, lemon juice, kosher salt, and pepper to broth. Bring to a boil over medium-high heat. Reduce heat to medium-low; simmer 8 minutes. Stir in reserved dill and parsley; cook 2 minutes. Garnish, if desired.